Magnetic susceptibility is a physical property intrinsic to a material and measures the amount of magnetization induced in that material when placed in an external magnetic field such as the main magnetic field, B0, of an MRI scanner. Using magnetic resonance imaging (MRI) to obtain a quantitative map of susceptibility provides a non-invasive means to measure the myelin in white matter tracts (see Li et al. Neuroimage 2011; 55, pp. 1645-1656; Liu. Magn Reson Med 2010; 63, pp. 1471-1477; Wu et al. Magn Reson Med 2012; 67, pp. 137-147), cortical gray matter (see Shmueli et al. Magn Reson Med 2009; 62, pp. 1510-1522), iron in deoxygenated or degraded blood and non-haem iron deposition (see Haacke et al. J Magn Reson Imaging 2010; 32, pp, 663-676; Liu et al. Magn Reson Med 2011; 66, pp, 777-783; Liu et al. Magn Reson Med 2009; 61, pp. 196-204; Schweser et al. Neuroimage 2011; 54, pp. 2789-2807; Wharton et al. Magn Reson Med 2010; 63, pp. 1292-1304; Wharton and Bowtell. Neuroimage 2010; 53, pp. 515-525), and the biodistribution of contrast agents for clinical and preclinical investigations (see Liu. Magn Reson Med 2010; 63, pp. 1471-1477; de Rochefort et al. Magn Reson Med 2008; 60, pp. 1003-1009; de Rochefort et al. lvled Phys 2008; 35, pp. 5328-5339; Liu et al. Magn Reson Imaging 2010; 28, pp. 1383-1389). Because of this promising potential, quantitative susceptibility mapping (QSM) has received increasing scientific and clinical attentions.
MRI Signal Model for Tissue Magnetic Susceptibility
The magnetic field controlling spin precession procedure can determine the MRI signal phase, allowing estimation of tissue local magnetic field (see Li. Magn Reson Med 2001; 46, pp, 907-916; Li and Leigh, Magnetic Resonance in Medicine 2004; 51, pp. 1077-1082; Li. American Journal of Physics 2002; 70, pp. 1029-1033; Li and Leigh. J Magn Reson 2001; 148, pp. 442-448). Tissue local magnetic field bL relative to, scaled to and along the main magnetic field B0 in image space (referred to as r-space) can be modeled as the convolution of the dipole kernel d(r)=(¼π)(3 cos2 θ−1)/|r|3 with tissue volumetric susceptibility distribution χ(r) plus observation noise n(r):bL(r)=d(r)χ(r)+n(r).  [0.1]
This signal model for tissue magnetic susceptibility can also be expressed in the Fourier dual k-space in matrix form:Bl(k)=D(k)X(k)+N(k)  [0.2]where Bl=FbL with F the Fourier transform operator, X=Fχ, and diagonal matrix D=Fd=⅓−kz2/k2, and N=Fn (see Koch et al. Phys Med Biol 2006; 51, pp. 6381-6402; Marques and Bowtell. Concepts in Magnetic Resonance Part B-Magnetic Resonance Engineering 2005; 25B, pp. 65-78; Salomir et al. Concepts in Magnetic Resonance Part B-Magnetic Resonance Engineering 2003; 19B, pp. 26-34).
The dipole kernel can have a non-trivial null space where D=0, which consists of a pair of opposing cone surfaces at the magic angle)(54.7°) with respect to the B0 direction. This null space can make the dipole kernel undersample or subsample the susceptibility, in addition to that the dipole kernel intertwines the susceptibility. Therefore, the inverse problem of quantitatively determining the susceptibility map (QSM) from the local field is fundamentally ill-posed, and can require additional information to uniquely determine the susceptibility. The mathematical nature of QSM, as expressed in Eq. 0.1 or Eq. 0.2, can be identical to that in super-resolution image reconstruction where input image data are subsampled, warped and blurred, and noisy (see Park et al. IEEE Signal Processing Magazine 2003; 20, pp. 16; Farsiu et al. IEEE transactions on image processing: a publication of the IEEE Signal Processing Society 2004; 13, pp. 1327-1344). Regularization methods using prior information, particularly these from the Bayesian stochastic approach, can be used to derive a unique estimate of the susceptibility. However, rigorous justification and evaluation are necessary to validate any regularization method.
The noise in the local field estimated from the MRI measured signal phase in general has a complex probability distribution that can be approximated as Gaussian only when there is MRI signal-to-noise ratio (SNR)≦≦1 (see Gudbjartsson and Patz. Magn Reson Med 1995; 34, pp. 910-914). The Gaussian approximation is typically used in literature because of its simplicity in formulation. However, the Gaussian approximation is not valid when SNR is poor at locations with strong susceptibilities. The phase noise variance in the Gaussian approximation is the square of the inverse of SNR and can vary in space in MRI. When using the Bayesian approach to formulate the data fidelity term, this spatially varying noise variance may need to be accounted for.
Previous Methods for Quantitative Susceptibility Mapping (QSM)
The mathematical nature for the inverse problem of determining susceptibility map from field measurement is a classical super-resolution (SR) image reconstruction problem (see Park et al. IEEE Signal Processing Magazine 2003; 20, pp. 16). Known SR methods can be employed for QSM reconstruction problem, including the Bayesian statistical approach by maximizing a posteriori, i.e., maximizing the posterior distribution (see Park et al. IEEE Signal Processing Magazine 2003; 20, pp. 16; Elad and Feuer. IEEE transactions on image processing: a publication of the IEEE Signal Processing Society 1997; 6, pp. 1646-1658) that readily gives a general solution for the susceptibility by minimizing a cost function consisting of a data fidelity term L[χ(r)] associated with the likelihood function and a regularization term R[χ(r)] associated with the prior distribution:χ(r)=argminχ(r)L[χ(r)]+αR[χ(r)],  [0.3]where L[χ(r)]=Σr|w(r)[bL(r)−d(r)χ(r)]|2 with w=SNR to account for the spatial variation of noise variance in the Gaussian approximation of MRI signal phase; the regularization term is usually expressed as the pth power of the Lp norm of a penalty function C that exponentially favors a solution of the desired property R[χ(r)]=∥C[χ(r)]∥pp. The anatomic prior information can be incorporated into the regularization term as a mask (see Gindi et al, Ieee Transactions on Medical Imaging 1993; 12, pp. 670-680) or weighting gradient (see Elad and Feuer. IEEE transactions on image processing: a publication of the IEEE Signal Processing Society 1997; 6, pp. 1646-1658; Baillet and Garnero. IEEE transactions on bio-medical engineering 1997; 44, pp. 374-385). Previous QSM work has contributed to the formulation of the regularization term R. Various regularizations have been tried, including the L1 norm and the square of the L2 norm of the gradient function (G) (see Kressler et al. Ieee Transactions on Medical Imaging 2010; 29, pp. 273-281), and the L1 norm of a wavelet-decomposition (see Wu et al. Magn Reson Med 2012; 67, pp. 137-147). One exemplary procedure can be the morphology enabled dipole inversion (MEDI) procedure (see Liu et al. Neuroimage 2012; 59, pp. 2560-2568) that includes an edge mask M from the gradient of anatomical prior a(r) such as the magnitude image in the regularization term, M(r)=1 for |Ga(r)|< threshold, M=0 otherwise, After vectorization, the MEDI regularization term isR=∥MGχ∥1.  [0.4]
There are also non-Bayesian methods, including the truncated k-space division (TKD) method in which a threshold value can be used as the dipole kernel when its absolute value is less than the threshold (see Shmueli et al. Magn Reson Med 2009; 62, pp. 1510-1522) and the weighted k-space derivative (WKD) method in which a linear interpolation of the dipole kernel is used when its absolute value is less than a threshold(see Li et al. Neuroimage 2011; 55, pp. 1645-1656). These division methods can suffer from severe noise amplifications in the region where the dipole kernel is small.
Exemplary Limitations of Previous Methods
There are certain major exemplary limitations in the previous methods that are addressed in the present disclosure. One exemplary limitation is that the anatomic prior information is formulated as a crude edge mask as in Eq. 0.4, which tends to make the reconstructed susceptibility map blocky or pixelated in the L1 norm. This can be because small details with low contrast to noise ratio (CNR) are not captured in Eq. 0.4.
Another exemplary limitation is that in regions of high susceptibility, the SNR may be very poor due to signal loss associated with large dephasing caused by susceptibility inhomogeneity. The Gaussian model for MRI signal phase can become highly erroneous, and consequently the data fidelity term in Eq. 0.3 causes severe artifacts in the reconstructed susceptibility map.
Accordingly, there is a need to address at least some of these exemplary deficiencies and/or limitations using the exemplary embodiments of the systems, process and computer-accessible medium according to exemplary embodiments of the present disclosure.